Given the volume, find the length of cuboids and the volume of a triangular prism. Also, calculate the surface area, area of compound shapes, area of parallelograms and trapeziums,... Given the volume, find the length of cuboids and the volume of a triangular prism. Also, calculate the surface area, area of compound shapes, area of parallelograms and trapeziums, area of triangles, and conversions between units. Can you also use diagrams?
Understand the Problem
The question is asking to find the length of cuboids given their volume, as well as to calculate the volume of a triangular prism. Additionally, it mentions related concepts such as surface area for cuboids, areas for various shapes (compound shapes, parallelograms, trapeziums, and triangles), and converting units. The user also requests the use of diagrams for clarification.
Answer
Length of cuboid: $6 \, \text{cm}$; Volume of triangular prism: $75 \, \text{cm}^3$.
Answer for screen readers
The length of the cuboid is ( 6 , \text{cm} ) and the volume of the triangular prism is ( 75 , \text{cm}^3 ).
Steps to Solve
- Calculate the Length of Cuboid from Volume
To find the length ( l ) of a cuboid, use the formula for the volume of a cuboid: $$ V = l \times w \times h $$ Rearranging the formula to solve for ( l ): $$ l = \frac{V}{w \times h} $$
- Finding the Volume of a Triangular Prism
The formula for the volume ( V ) of a triangular prism is: $$ V = \text{Base Area} \times \text{Height} $$ Calculate the base area using the formula for the area of triangle: $$ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height}_{\text{triangle}} $$
Combine both formulas to find the volume of the triangular prism: $$ V = \left( \frac{1}{2} \times \text{base} \times \text{height}{\text{triangle}} \right) \times h{\text{prism}} $$
- Example Values for Calculation
Assume for the cuboid:
- Volume ( V = 240 , \text{cm}^3 )
- Width ( w = 10 , \text{cm} )
- Height ( h = 4 , \text{cm} )
Plug in the values: $$ l = \frac{240}{10 \times 4} = \frac{240}{40} = 6 , \text{cm} $$
For the triangular prism:
- Base ( = 5 , \text{cm} )
- Height of triangle ( = 3 , \text{cm} )
- Height of prism ( = 10 , \text{cm} )
Calculate base area: $$ \text{Base Area} = \frac{1}{2} \times 5 \times 3 = 7.5 , \text{cm}^2 $$ Now find the volume: $$ V = 7.5 \times 10 = 75 , \text{cm}^3 $$
The length of the cuboid is ( 6 , \text{cm} ) and the volume of the triangular prism is ( 75 , \text{cm}^3 ).
More Information
This exercise involves basic geometric calculations. Understanding how to manipulate volume formulas for different shapes helps in both practical applications and theoretical learning in geometry.
Tips
- Confusing the dimensions when identifying width, height, or length.
- Incorrectly substituting values into the volume formulas.
- Forgetting to use consistent units throughout the calculation.
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